Michael Corral: Vector Calculus

12 CHAPTER 1. VECTORS IN EUCLIDEAN SPACE
Let u=(u 1 ,u 2 ,u 3 ), v=(v 1 ,v 2 ,v 3 ), w=(w 1 ,w 2 ,w 3 ) be vectors in 3 . Then
u+(v+w)=(u 1 ,u 2 ,u 3 )+((v 1 ,v 2 ,v 3 )+(w 1 ,w 2 ,w 3 ))
= (u 1 ,u 2 ,u 3 )+(v 1 +w 1 ,v 2 +w 2 ,v 3 +w 3 ) by Theorem 1.4(b)
= (u 1 +(v 1 +w 1 ),u 2 +(v 2 +w 2 ),u 3 +(v 3 +w 3 )) by Theorem 1.4(b)
= ((u 1 +v 1 )+w 1 ,(u 2 +v 2 )+w 2 ,(u 3 +v 3 )+w 3 ) by properties of real numbers
= (u 1 +v 1 ,u 2 +v 2 ,u 3 +v 3 )+(w 1 ,w 2 ,w 3 ) by Theorem 1.4(b)
= (u+v)+w
This completes the analytic proof of (b). Figure 1.2.6 provides the geometric proof.
u+(v+w)=(u+v)+w
u
u+v
v+w
w
v
Figure 1.2.6 Associative Law for vector addition
(c) We already discussed this on p.10.
(d) We already discussed this on p.10.
(e) We will prove this for a vector v=(v 1 ,v 2 ,v 3 ) in 3 (the proof for 2 is similar):
k(lv)=k(lv 1 ,lv 2 ,lv 3 )
by Theorem 1.4(a)
= (klv 1 ,klv 2 ,klv 3 ) by Theorem 1.4(a)
= (kl)(v 1 ,v 2 ,v 3 ) by Theorem 1.4(a)
= (kl)v
(f) and (g): Left as exercises for the reader.
QED
A unit vector is a vector with magnitude 1. Notice that for any nonzero vector v,
the vector v
1
‖v‖
is a unit vector which points in the same direction as v, since
‖v‖ > 0
and ∥ ∥
v∥
‖v‖
∥= ‖v‖
‖v‖
= 1. Dividing a nonzero vector v by‖v‖ is often called normalizing v.
There are specific unit vectors which we will often use, called the basis vectors:
i=(1,0,0), j=(0,1,0), and k=(0,0,1) in 3 ; i=(1,0) and j=(0,1) in 2 .
These are useful for several reasons: they are mutually perpendicular, since they lie
on distinct coordinate axes; they are all unit vectors:‖i‖=‖j‖=‖k‖=1; every vector
can be written as a unique scalar combination of the basis vectors: v=(a,b)=ai+bj
in 2 , v=(a,b,c)=ai+bj+ck in 3 . See Figure 1.2.7.